Improved detection of magnetic interactions in proteins based on long-lived coherences

Living systems rely on molecular building blocks with low structural symmetry. Therefore, constituent amino acids and nucleotides yield short-lived nuclear magnetic responses to electromagnetic radiation. Magnetic signals are at the basis of molecular imaging, structure determination and interaction studies. In solution state, as the molecular weight of analytes increases, coherences with long lifetimes are needed to yield advantageous through-space magnetisation transfers. Interactions between magnetic nuclei can only be detected provided the lifetimes of spin order are sufficient. In J-coupled pairs of nuclei, long-lived coherences (LLC’s) connect states with different spin-permutation symmetry. Here in, we show sustained LLC’s in protein Lysozyme, weighing 14.3 kDa, with lifetimes twice as long as those of classical magnetisation for the aliphatic protons of glycine residues. We found for the first time that, in a protein of significant molecular weight, LLC’s yield substantial through-space magnetisation transfers: spin-order transfer stemming from LLC’s overcame transfers from classical coherences by factors > 2. Furthermore, in agreement with theory, the permutation symmetry of LLC-based transfers allows mapping interacting atoms in the protein structure with respect to the molecular plane of glycine residues in a stereospecific manner. These findings can extend the scope of liquid-state high-resolution biomolecular spectroscopy.


Supplementary Note 1: Experimental workflow for selective long-lived coherence experiments in large proteins
Step 1: Broadband excitation and detection of LLS via pulse sequence of Ref. [3].Glycine resonances are identified (blue) in this way, but are not clearly assigned.
Step 2: Based on knowledge of resonance frequencies acquired at Step 1, selective excitation and detection of Gly49 signal (green -LLS-relaxation filtered) was detected using the frequency-selective pulse sequence of Ref. [4].This method allows for unambiguous assignment of coupled spins resonances in each glycine using an LLS filter.Step 3. Once both resonances of one glycine residue are assigned, a 50 ms polychromatic EBURP2 pulse (with initial frequency offset) allows simultaneous excitation of transverse magnetization in two spectral regions with opposite sign, leading to LLC excitation of the selected Glycine residue (red).The lower spectrum shows the experimental profile of the selective pulse in the protein sample.The excitation spectral regions (with a spectral width around 70 Hz each) are centered on the previously assigned glycine resonances.2. Supplementary Note 2: Theoretical analysis of rotating-frame Overhauser transfer from long-lived states and coherences In the following we consider the evolution of a system of three spins, I, S, and K, with relaxation due to dipolar interactions only, and J-coupling between I and S. For Overhauser magnetization transfer starting from long-lives states, we note that the cross-relaxation rate constant between   =  ̂•  ̂ and another 1-spin order term of the third spin  ̂, with  = , , , is null ( /  = 0) when only dipolar interactions are considered.This can be understood in terms of operators' mathematical properties as follows: because dipolar interactions are characterized by 2-spin order Hamiltonians  ̂ =   (3 ̂ ̂ −  ̂•  ̂), the double-commutation superoperator  ̂ = ⟦ ̂ , [ ̂ , ⟧ can only modify the correlation order of a targeted operator with 0 or ± 2, as discussed more thoroughly in Ref. [1].Thus, the singlet population   =  ̂•  ̂ cannot be converted directly through Overhauser effect into a 1-spin term such as polarization at a third neighboring spin.Nonetheless, if cross-correlated effects between the dipolar mechanism and other relaxation interactions characterized by 1-spin order Hamiltonians are present, the crossrelaxation rate  /  will become non-null and polarization transfer can take place.Chemical shift anisotropy (CSA) is such a mechanism that should allow polarization transfer from long-lived states based on heavier ½-spin nuclei, such as 13 C, 19 F and 31 P, where the shielding anisotropy is large.Additional analytical investigation was performed within SpinDynamica [2].
For the case of long-lived coherences (LLC's), discussed in the main text: The expression of long-lived coherences in terms of Cartesian component can be expressed, in the habitual z-axis quantization given by an external B0 field, as a real component   (−) // =   −   and one imaginary component with zero-quantum terms,  (−) ┴ = (2    − 2    ).In the presence of a continuous-wave sustaining field B1 applied along the x-axis of the laboratory frame, with amplitude superior to the frequency difference between the two spins, the quantization axis changes from the z to the x axis, yielding: Their time evolution is given by: The operator    =  ̂ −  ̂ is a linear combination of 1-spin order terms and has a non-zero cross-relaxation rate constant with the third spin operator  ̂ equal to the difference between individual cross-relaxation rates of the two I and S spins with the third spin K (   // /  =    /  −    /  ).This leads to a pronounced angular dependence of the LLC-ROE transfer with a maximum cross-relaxation rate when the three spins are collinear, and a minimum value when the third spin K is on the perpendicular bisector of the segment connecting I and S nuclei.The operator    = 2 ̂ ̂ − 2 ̂ ̂ is a sum of 2-spin order terms and, based on the above discussion, has a null cross-relaxation rate with any form of polarization on the third spin  ̂, with  = , ,  (   ⊥ /  = 0).Because long-lived coherences oscillate between the two components (   and    ) with a frequency equal to the scalar coupling constant   , the ROE transfer towards the  ̂ obtained from spin-order initially excited via    will oscillate with the same frequency, but with a phase difference of /2 with respect to the LLC's evolution.
With increasing molecular weight or solvent viscosity, the characteristic rotational correlation times of analyte molecules,   , increase.LLC's lifetimes become shorter than one oscillation period and the ROE transfer is described by a non-oscillating bi-exponential curve.Nonetheless, due to the longer lifetime of LLC compared to classical coherences (up to 9 times higher) [3], an enhanced ROE transfer is predicted by numerical simulations for large proteins [4].
From Eq. [1-3], the evolution of    and    is given by ⁄ = +2   −      [6] where   =   −   and    =   −   =  − .The equation for the change rate of ,   ⁄ = −   −    −   , [7] can be recast as where  + =   +   .Because of the appearance of  + in Eq. [7], an additional equation must be added which, according to Eq. [1] and [2], is where  + =   +   .However, according to the experimental procedure for producing LLC, initially  + (0) = 〈  +   〉(0) = 0 and it can be expected that  + () remains small throughout the ROESY irradiation.Therefore, we can ignore Eq. [8] and neglect  +  + term in Eq. [7], such that we are left with the system of equations  Predicted oscillations at the source frequency (Gly-2 JIS = 17 Hz) in the LLC-transferred magnetization can hamper the magnetization build-up, but are in practice quenched over large domains of the observed times by oscillations of the detected spins K, explaining the experimentally-detected rotating-frame Overhauser transfers.

Supplementary Note 3.2: Matlab notebooks for calculation of LLC-ROE intensities
Rotating-frame Overhauser transfer intensities in presence of additional external spins

Supplementary Note 3.3: Build-up in NOE vs ROE and ROE_LLC spectroscopy
The ratio of the enhancements for NOE and ROE transfer for a large molecule (ω 0 τ  ≫ 0), is The pertaining calculations [8,9] are detailed here.
In a homonuclear system: In the slow-diffusion limit (ω 0 τ  ≫ 0), this value becomes: The value of  1 can be computed as: where Regarding the ROE, we have in the slow-diffusion limit: The transverse relaxation rate constant in the rotating frame is: In the same slow-tumbling limit, this becomes: Given that we have experimentally obtained values of η   up to two times larger than "standard" η  measurements, we can reasonably expect this method to also provide an improvement over the NOE measurement.

Figure S1 .
Figure S1.Workflow employed for the identification of glycine resonances and selective LLC excitation of glycine residues in Lysozyme at 950 MHz.Step 1: Broadband excitation and detection of LLS via pulse sequence of Ref. [3].Glycine resonances are identified (blue) in this way, but are not clearly assigned.Step 2: Based on knowledge of resonance frequencies acquired at Step 1, selective excitation and detection of Gly49 signal (green -LLS-relaxation filtered) was detected using the frequency-selective pulse sequence of Ref. [4].This method allows for unambiguous assignment of coupled spins resonances in each glycine using an LLS filter.Step 3. Once both resonances of one glycine residue are assigned, a 50 ms polychromatic EBURP2 pulse (with initial frequency offset) allows simultaneous excitation of transverse magnetization in two spectral regions with opposite sign, leading to LLC excitation of the selected Glycine residue (red).The lower spectrum shows the experimental profile of the selective pulse in the protein sample.The excitation spectral regions (with a spectral width around 70 Hz each) are centered on the previously assigned glycine resonances.

3. Supplementary Note 3 : 3 . 1 Supplementary Note 3 . 1 :
Figure S2.Simulated angular dependence of the magnetization transfer from ROE LLC as function of  angle at   = 15  rotational correlation time.The {, , } spin system has the following geometrical constrains: the internuclear distance between I and S spins is 1.76 Å and the distance between K spin and the middle point of the I-S segment is d = 4.0 Å (Spinach [7] notebook provided as supplementary materials).Stereospecific ROE transfer is obtained from LLC's by sign reversal of the build-up due to a closer proximity of the K spin toward either I or S spins.

Figure
Figure S5 A) (two-spins systems) Comparison of Overhauser transfer   in 2-spin systems according to calculations above, plotted as in H. Desvaux, P. Berthault, N. Birlirakis, M. Goldman, "Off-Resonance ROESY for the study of dynamic processes", J. Magn.Reson.Ser. A. 108 (1994) 219 229; B) (three-spins systems) Comparison of ROE-LLC and standard ROE transfer between a two-spin source and a third spin.(Spinach simulation).

Figure
Figure S6 (top) Source LLC for Gly-117 in 2D spectroscopy.(bottom) ROE LLC 2D spectrum (recorded with pulse sequence from Fig 1C) superimposed with slices from ROESY 2D spectrum for Gly117 (in green -1D slice at H α2 frequency, in blue -1D slice at H α3 frequency).Spatial neighbors within a 10 Å radius have been assigned based on the 9LYZ PDB structure.Strong signals that could not be assigned with certainty are likely to originate from: Trp111(HD1, HH2, HE3) in the 6.5-7.5 ppm region, respectively T118, V120 HG in the 1-2 ppm region.

F 2 Figure S7 .
Figure S7.Interaction of trisaccharide NAM-NAG-NAM (blue) with Lysozyme based on PDB structure '9LYZ'.Long-lived coherences excited on aliphatic protons of widespread Gly residues can sense biological interactions directly or via Overhauser-interaction mediated transfers (e.g., Gly49H α2,3 interact with Asn46H α , which is in direct contact with the ligand).

Table S1 .
Assigned glycine resonances using the workflow method described above The accuracy of Eq. [14] was tested for various values of the relevant parameters.We have found that Eq. [14] is accurate for realistic values of   ,   , and . Significant deviations between () and numerically-computed values occur only when    is comparable to   (unlikely, as internuclear distances      are in all cases larger than   ).